2 Answers
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Baby tries to use the bottle-feeding technique on the breast and has difficulty latching-on and sucking. Baby gets very frustrated, and so does mother. Nipple confusion can even lead to baby refusing the breast. Here's an explanation.

Not all children get it (mine didn't), but I've had friends whose kids have definitely had issues with this. This article discusses the timing of introduction of the pacifier and the bottle. The article is lengthy, and also discusses bottle feeding vs. breast feeding (germane, as the problem is nipple confusion, which can arise here as well). Her final recommendations with bottles (unless you have to supplement feeding) are:

At 3-4 weeks (or 2-3 weeks if you are returning to work at 6 weeks), begin pumping after feedings for 4-5 minutes if your baby had a good feeding, or 8-10 minutes if he didn’t nurse well or only nursed on one breast.

EDIT: Also, in case you're worried about potential effects later in life (ie, the possible need for braces caused or exacerbated by the use of pacifiers or bottle feeding), there does not appear to be a statistically relevant link, according to this study. From the abstract:

Analysis of valid responses (N = 454) showed need for treatment to be associated only with history of parental orthodontic treatment (P < 0.05). A trend toward association of bottle-feeding with need for orthodontic treatment was found in the increasing need for treatment with increasing exposure to bottle, but was of marginal significance (P = 0.058).

Your pediatrician will keep an eye on the baby's physical development in their mouth, and should let you know when they've been using the pacifier too long.

When babies can learn to self-sooth, they are a lot of the way along to learning a skill that's vital for falling back to sleep without help. Even if they don't always use the pacifier at bedtime, they will have learned that they don't always need your help to calm down. Yay!

This is correct, but do note that if they don't use a pacifier, but their thumb instead, this has another drawback: a pacifier can be "given to Santa" - a thumb cannot.
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KonerakMay 8 '12 at 8:20

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None of the pediatricians I've taken my kids to have ever even broached the discussion of pacifier usage. Granted, we got rid of my son's pacifiers when he was 2 1/2 and will probably do something similar with my daughter when she gets older. Just don't rely on your ped to say anything about your child's oral development--some won't until real damage has all ready occurred.
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Meg CoatesMay 8 '12 at 15:17

Mag, this might vary among pediatricians. Ours specifically discussed it with us, suggesting that using a pacifier for too long could cause problems like overbite, etc. -- but she didn't make a big deal about it. (After long experience, we trust her to speak up when she sees something wrong, and she trusts me to ask TONS of dumb questions.)
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Will E.May 22 '12 at 18:12